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Minigame probabilities help

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GAThraawn
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Joined: 01/22/2013

I'm working on a simple minigame to be a small part of a larger game that I'm designing, and I want to be able to tweak aspects of it to modify the probabilities of winning, but not being terribly good with mathematics I've been brute forcing the probabilities, which has been time consuming and possibly error filled. I would be grateful if anyone with a better head for these things could help me.

This is the game:

One player is designated the attacker, and another the defender. Each player is given two 2-value cards and two 3-value cards, and secretly chooses two of their cards to play face down. They are then flipped face up and each card of the attacker's that matches one of the defender's is discarded, and the attacker then scores the remainder in points.
So for example if the attacker places 3/3, he will score 0, 3 or 6 points, depending on if the defender played 3/3, 3/2 or 2/2.
This is repeated twice more, and by the end of the third round, if the attacker has 8 or more points he wins, otherwise the defender wins.

My current estimation of the probabilites, accounting for optimal play (if the defender plays 2/3, he can guarantee the attacker scores no more than 3 points each round, which for example makes winning impossible if the attacker scored 0 in the first round) is that the attacker has a 44% chance to win if they start with 3/3, a 27% chance if they start with 2/2 and a 24% chance if they start with 3/2.

I'm curious how this game changes with modifications such as: reducing the number of points needed to win to 7, or even 6, or guaranteeing the attacker gets at least 1 point each round. Particularly, I would like to know how often the attacker is likely to win with any of these variants, as currently it seems that the attacker has less than 50% odds of winning.

If anyone has any suggestions or wants to offer some help with the numbers, it would be greatly appreciated. I am aware that this is not going to be a terribly interesting game in a vaccuum, but it serves its function as a simple sub-system. It doesn't need to become a perfectly fair game, I just want to increase the attacker's chances, and maybe have some bonuses that could be obtained in the rest of the game to make it easier.

Thanks for the help.

Orangebeard
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Joined: 10/13/2011
Exact Match?

I need to work through the statistics, but would it still work if the defender had to match each card exactly? For example, the attacker plays a 2 in position 1 and a 3 in position 2; the defender plays a 3 in position 1 and a 2 in position 2; the attacker does 5 damage since neither card is an exact match.

GAThraawn
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Joined: 01/22/2013
My first thought was that

My first thought was that that would introduce too much random guessing to an already random guesswork system, but upon reflection I suppose it would only even matter if each player played 3/2, which is already a fairly weak strategy, so sure, I'd be interested in seeing those numbers. Thanks for the suggestion.
I also cross posted on BGG, where it was suggested that one of the fairest distributions would be to have four rounds and play to 9 points, although that still gives the attacker a less than 50% chance. I wouldn't mind coming up with some options that give the attacker a better than 50% chance.

Thanks for your help.

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